Consider an $\mathrm{M} / \mathrm{G} / r / 0$ loss system with arrival rate $\lambda$ and service-time distribution $F$. Thus, arrivals form a Poisson process of rate $\lambda$, service times are independent with common distribution $F$, there are $r$ servers and there is no space for waiting. Use Little's Lemma to obtain a relation between the long-run average occupancy $L$ and the stationary probability $\pi$ that the system is full.

Cafe-Bar Duo has 23 serving tables. Each table can be occupied either by one person or two. Customers arrive either singly or in a pair; if a table is empty they are seated and served immediately, otherwise, they leave. The times between arrivals are independent exponential random variables of mean $20 / 3$. Each arrival is twice as likely to be a single person as a pair. A single customer stays for an exponential time of mean 20 , whereas a pair stays for an exponential time of mean 30 ; all these times are independent of each other and of the process of arrivals. The value of orders taken at each table is a constant multiple $2 / 5$ of the time that it is occupied.

Express the long-run rate of revenue of the cafe as a function of the probability $\pi$ that an arriving customer or pair of customers finds the cafe full.

By imagining a cafe with infinitely many tables, show that $\pi \leqslant \mathbb{P}(N \geqslant 23)$ where $N$ is a Poisson random variable of parameter $7 / 2$. Deduce that $\pi$ is very small. [Credit will be given for any useful numerical estimate, an upper bound of $10^{-3}$ being sufficient for full credit.]